# Properties

 Label 950.2.a.h Level $950$ Weight $2$ Character orbit 950.a Self dual yes Analytic conductor $7.586$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(1,950)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + \beta q^{7} + q^{8} + (\beta + 1) q^{9}+O(q^{10})$$ q + q^2 + b * q^3 + q^4 + b * q^6 + b * q^7 + q^8 + (b + 1) * q^9 $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + \beta q^{7} + q^{8} + (\beta + 1) q^{9} + 4 q^{11} + \beta q^{12} + ( - 3 \beta + 2) q^{13} + \beta q^{14} + q^{16} + (\beta - 6) q^{17} + (\beta + 1) q^{18} - q^{19} + (\beta + 4) q^{21} + 4 q^{22} - 3 \beta q^{23} + \beta q^{24} + ( - 3 \beta + 2) q^{26} + ( - \beta + 4) q^{27} + \beta q^{28} + ( - 3 \beta + 2) q^{29} - 2 \beta q^{31} + q^{32} + 4 \beta q^{33} + (\beta - 6) q^{34} + (\beta + 1) q^{36} + 6 q^{37} - q^{38} + ( - \beta - 12) q^{39} + (4 \beta + 2) q^{41} + (\beta + 4) q^{42} + ( - 2 \beta + 8) q^{43} + 4 q^{44} - 3 \beta q^{46} + ( - 4 \beta + 4) q^{47} + \beta q^{48} + (\beta - 3) q^{49} + ( - 5 \beta + 4) q^{51} + ( - 3 \beta + 2) q^{52} + (\beta + 2) q^{53} + ( - \beta + 4) q^{54} + \beta q^{56} - \beta q^{57} + ( - 3 \beta + 2) q^{58} + \beta q^{59} + (2 \beta + 6) q^{61} - 2 \beta q^{62} + (2 \beta + 4) q^{63} + q^{64} + 4 \beta q^{66} + \beta q^{67} + (\beta - 6) q^{68} + ( - 3 \beta - 12) q^{69} + 4 \beta q^{71} + (\beta + 1) q^{72} + (3 \beta - 6) q^{73} + 6 q^{74} - q^{76} + 4 \beta q^{77} + ( - \beta - 12) q^{78} - 2 \beta q^{79} - 7 q^{81} + (4 \beta + 2) q^{82} + (2 \beta - 8) q^{83} + (\beta + 4) q^{84} + ( - 2 \beta + 8) q^{86} + ( - \beta - 12) q^{87} + 4 q^{88} + 2 q^{89} + ( - \beta - 12) q^{91} - 3 \beta q^{92} + ( - 2 \beta - 8) q^{93} + ( - 4 \beta + 4) q^{94} + \beta q^{96} - 6 q^{97} + (\beta - 3) q^{98} + (4 \beta + 4) q^{99} +O(q^{100})$$ q + q^2 + b * q^3 + q^4 + b * q^6 + b * q^7 + q^8 + (b + 1) * q^9 + 4 * q^11 + b * q^12 + (-3*b + 2) * q^13 + b * q^14 + q^16 + (b - 6) * q^17 + (b + 1) * q^18 - q^19 + (b + 4) * q^21 + 4 * q^22 - 3*b * q^23 + b * q^24 + (-3*b + 2) * q^26 + (-b + 4) * q^27 + b * q^28 + (-3*b + 2) * q^29 - 2*b * q^31 + q^32 + 4*b * q^33 + (b - 6) * q^34 + (b + 1) * q^36 + 6 * q^37 - q^38 + (-b - 12) * q^39 + (4*b + 2) * q^41 + (b + 4) * q^42 + (-2*b + 8) * q^43 + 4 * q^44 - 3*b * q^46 + (-4*b + 4) * q^47 + b * q^48 + (b - 3) * q^49 + (-5*b + 4) * q^51 + (-3*b + 2) * q^52 + (b + 2) * q^53 + (-b + 4) * q^54 + b * q^56 - b * q^57 + (-3*b + 2) * q^58 + b * q^59 + (2*b + 6) * q^61 - 2*b * q^62 + (2*b + 4) * q^63 + q^64 + 4*b * q^66 + b * q^67 + (b - 6) * q^68 + (-3*b - 12) * q^69 + 4*b * q^71 + (b + 1) * q^72 + (3*b - 6) * q^73 + 6 * q^74 - q^76 + 4*b * q^77 + (-b - 12) * q^78 - 2*b * q^79 - 7 * q^81 + (4*b + 2) * q^82 + (2*b - 8) * q^83 + (b + 4) * q^84 + (-2*b + 8) * q^86 + (-b - 12) * q^87 + 4 * q^88 + 2 * q^89 + (-b - 12) * q^91 - 3*b * q^92 + (-2*b - 8) * q^93 + (-4*b + 4) * q^94 + b * q^96 - 6 * q^97 + (b - 3) * q^98 + (4*b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 + q^7 + 2 * q^8 + 3 * q^9 $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9} + 8 q^{11} + q^{12} + q^{13} + q^{14} + 2 q^{16} - 11 q^{17} + 3 q^{18} - 2 q^{19} + 9 q^{21} + 8 q^{22} - 3 q^{23} + q^{24} + q^{26} + 7 q^{27} + q^{28} + q^{29} - 2 q^{31} + 2 q^{32} + 4 q^{33} - 11 q^{34} + 3 q^{36} + 12 q^{37} - 2 q^{38} - 25 q^{39} + 8 q^{41} + 9 q^{42} + 14 q^{43} + 8 q^{44} - 3 q^{46} + 4 q^{47} + q^{48} - 5 q^{49} + 3 q^{51} + q^{52} + 5 q^{53} + 7 q^{54} + q^{56} - q^{57} + q^{58} + q^{59} + 14 q^{61} - 2 q^{62} + 10 q^{63} + 2 q^{64} + 4 q^{66} + q^{67} - 11 q^{68} - 27 q^{69} + 4 q^{71} + 3 q^{72} - 9 q^{73} + 12 q^{74} - 2 q^{76} + 4 q^{77} - 25 q^{78} - 2 q^{79} - 14 q^{81} + 8 q^{82} - 14 q^{83} + 9 q^{84} + 14 q^{86} - 25 q^{87} + 8 q^{88} + 4 q^{89} - 25 q^{91} - 3 q^{92} - 18 q^{93} + 4 q^{94} + q^{96} - 12 q^{97} - 5 q^{98} + 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 + q^7 + 2 * q^8 + 3 * q^9 + 8 * q^11 + q^12 + q^13 + q^14 + 2 * q^16 - 11 * q^17 + 3 * q^18 - 2 * q^19 + 9 * q^21 + 8 * q^22 - 3 * q^23 + q^24 + q^26 + 7 * q^27 + q^28 + q^29 - 2 * q^31 + 2 * q^32 + 4 * q^33 - 11 * q^34 + 3 * q^36 + 12 * q^37 - 2 * q^38 - 25 * q^39 + 8 * q^41 + 9 * q^42 + 14 * q^43 + 8 * q^44 - 3 * q^46 + 4 * q^47 + q^48 - 5 * q^49 + 3 * q^51 + q^52 + 5 * q^53 + 7 * q^54 + q^56 - q^57 + q^58 + q^59 + 14 * q^61 - 2 * q^62 + 10 * q^63 + 2 * q^64 + 4 * q^66 + q^67 - 11 * q^68 - 27 * q^69 + 4 * q^71 + 3 * q^72 - 9 * q^73 + 12 * q^74 - 2 * q^76 + 4 * q^77 - 25 * q^78 - 2 * q^79 - 14 * q^81 + 8 * q^82 - 14 * q^83 + 9 * q^84 + 14 * q^86 - 25 * q^87 + 8 * q^88 + 4 * q^89 - 25 * q^91 - 3 * q^92 - 18 * q^93 + 4 * q^94 + q^96 - 12 * q^97 - 5 * q^98 + 12 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.56155 2.56155
1.00000 −1.56155 1.00000 0 −1.56155 −1.56155 1.00000 −0.561553 0
1.2 1.00000 2.56155 1.00000 0 2.56155 2.56155 1.00000 3.56155 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.a.h 2
3.b odd 2 1 8550.2.a.br 2
4.b odd 2 1 7600.2.a.y 2
5.b even 2 1 190.2.a.d 2
5.c odd 4 2 950.2.b.f 4
15.d odd 2 1 1710.2.a.w 2
20.d odd 2 1 1520.2.a.n 2
35.c odd 2 1 9310.2.a.bc 2
40.e odd 2 1 6080.2.a.bb 2
40.f even 2 1 6080.2.a.bh 2
95.d odd 2 1 3610.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 5.b even 2 1
950.2.a.h 2 1.a even 1 1 trivial
950.2.b.f 4 5.c odd 4 2
1520.2.a.n 2 20.d odd 2 1
1710.2.a.w 2 15.d odd 2 1
3610.2.a.t 2 95.d odd 2 1
6080.2.a.bb 2 40.e odd 2 1
6080.2.a.bh 2 40.f even 2 1
7600.2.a.y 2 4.b odd 2 1
8550.2.a.br 2 3.b odd 2 1
9310.2.a.bc 2 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(950))$$:

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - T - 38$$
$17$ $$T^{2} + 11T + 26$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 3T - 36$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} + 2T - 16$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} - 8T - 52$$
$43$ $$T^{2} - 14T + 32$$
$47$ $$T^{2} - 4T - 64$$
$53$ $$T^{2} - 5T + 2$$
$59$ $$T^{2} - T - 4$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} - T - 4$$
$71$ $$T^{2} - 4T - 64$$
$73$ $$T^{2} + 9T - 18$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} + 14T + 32$$
$89$ $$(T - 2)^{2}$$
$97$ $$(T + 6)^{2}$$